4 1. Jacobi operators
Furthermore,
£Q(I,M)
denotes the set of sequences with only finitely many
values being nonzero and -^_(Z,M) denotes the set of sequences in ^(Z, M) which
are
£p
near ±00, respectively (i.e., sequences whose restriction to ^(±N, M) belongs
to £
P
(±N,M). Here N denotes the set of positive integers). Note that, according
to our definition, we have
(1.3) £0(I, M) C
£P(I,
M) C JP(I, M) C e°°(I, M), p q,
with equality holding if and only if / is finite (assuming dimM 0).
In addition, if M is a (separable) Hilbert space with scalar product (.,..)M,
then the same is true for
£2(I,
M) with scalar product and norm defined by
n£l
(1-4) 11/1 1 = II/II2 = v ^ 7 . f,gef(I,M).
For what follows we will choose / = Z for simplicity. However, straightforward
modifications can be made to accommodate the general case / C Z.
During most of our investigations we will be concerned with difference expres-
sions, that is, endomorphisms of £(Z);
R: t(Z) -+ £(Z)
U'5)
f » Rf
(we reserve the name difference operator for difference expressions defined on a
subset of
£2(Z)).
Any difference expression R is uniquely determined by its corre-
sponding matrix representation
(1.6) ( # ( r a , n ) )
m n e Z
, R(m,ri) = (RSn)(m) = (8m,R8n),
where
(1.7) 6n{m) = 6min = i
1 n = m
is the canonical basis of £(Z). The order of R is the smallest nonnegative integer
N = N++N_ such that R{m, n) = 0 for all m, n with n m N+ and m n 7V_.
If no such number exists, the order is infinite.
We call R symmetric (resp. skew-symmetric) if R(m, n) = R(n, m) (resp.
R{m,n) = —R(n,m)).
Maybe the simplest examples for a difference expression are the shift expres-
sions
(1.8)
(S±f)(n)=f(n±l).
They are of particular importance due to the fact that their powers form a basis
for the space of all difference expressions (viewed as a module over the ring £{T)).
Indeed, we have
(1.9) i? = £ ( . , . +
fc)(S+)fe,
( 5
±
) "
1
= 5
T
.
kez
Here i?(.,. + k) denotes the multiplication expression with the sequence (R(n,n +
k))n£Zi that is, i?(.,. + k) : f(n) i-» R(n,n + k)f(n). In order to simplify notation
we agree to use the short cuts
(1.10)
f±=S±f, f++=S+S+f,
etc.,
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